Wave Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 52

the reflected waves appear. The interaction between the waves and the reflected
waves forms standing waves. For any individual standing wave, the displacement
can be written in the form of the product of function X ( x ) of the spatial position x

and function T of the time t . At any time instant, the string displacement is the

sum of that of all standing waves, i.e.

( t )

= ∑ k X k ( x ) T k ( t ) . u

The Fourier method in Section 2.2.1 uses the characteristics of the boundary con-
ditions to fix the type of X k ( , substitutes the series solution into the equation of

the PDS, and applies the initial conditions to determine T k (

x )

. Finally, we arrive at
Eq.(2.9) in Section 2.2.1, the series solution of separable variables, where each term
in the series satisfies the boundary conditions. Mathematically, it is a superposition
to express the solution in a form of series. Note that the PDS indeed satisfies the
conditions for applying the principle of superposition.
To apply the method of separation of variables to solve PDS (2.15), assume a
solution of type

t )

X (

( , )= ( ) ( ) , ( )= , )=

u x t X x T t X 0 0 l 0

and substituting it to the equation of PDS (2.15) to obtain

X ( T ( x ) t )

) = ) , a 2 T

X ( x ( t

where the primes on the functions X and T represent differentiation with respect
to the only variable present. Now the left-hand side of the above equation is inde-
pendent of t and the right-hand side is independent of x . Since they are equal, their
common value cannot be a function of x or t , and must be a constant, say − λ called

separation constant . Thus

X + λ X (

X = 0 , X ( 0 )= 0 , l )= 0 . (2.16)

T + λ a 2 T = 0 . (2.17)

The partial differential equation of PDS (2.15) has thus been reduced to two ordinary
differential equations.
The auxiliary problem defined by Eq.(2.16) is called an eigenvalue problem ,be-
cause it has solutions only for certain values of the separation constant

λ = λ k ,

which are called the eigenvalues ; the corresponding solution X k ( x ) are called the

eigenfunctions of the problem. When λ is not an eigenvalue, that is, when λ = λ k ,

the problem has trivial solutions (i.e. X λ = λ k ).

The eigenvalue problem given by Eq.(2.16) is a special case of a more general
eigenvalue problem called the Sturm-Liouville problem ,the S-L problem for short.
A discussion of the Sturm-Liouville problem is available in Appendix D.

= 0if